# Notes for Measure Theory

I am not a mathematician, my only higher level mathematics course was, what would now be called, a service course. I took this course over 40 years ago and the approach was very practical. So as I work through the book on measure theory I am having to look up definitions which would probably be obvious to a newer mathematics graduate.

The Axiom of Choice – if a set ${A}$ is partitioned into a number of non-empty disjoint subsets, ${P_i}$, the axiom of choice says that we can always obtain a set, ${T}$, by choosing one element from each of the ${P_i}$. This seems self evident when the number of subsets is finite; the axiom of choice extends this to the case where the number of subsets is infinite.
Countable – a set, ${A}$, is countable if one can define a bijective mapping from ${A}$ to a subset of the set of natural numbers, ${\mathbb{N}}$. If the mapping is from ${A}$ to ${\mathbb{N}}$ then ${A}$ is said to be countably infinite.
Infimum – given a set ${A}$, the infimum, ${\inf A}$, is the greatest lower bound of ${A}$. For example, if ${A=\{1/n: n\in \mathbb{N}\}}$, there is no minimum value, but ${\inf A}$ is defined and equals ${0}$. Note that ${\inf A \notin A}$,
Supremum – given a set ${A}$, the supremum, ${\sup A}$, is the least upper bound of ${A}$. For example, if ${A=\{1-1/n: n\in \mathbb{N}\}}$, there is no maximum value, however ${\sup A}$ is defined and equals ${1}$. Note that ${\sup A \notin A}$,
Uncountable – a set ${A}$ is uncountable if, and only if, it is not finite and not countable. The set of real numbers ${\mathbb{R}}$ is uncountable.